Homogeneous projective coordinates for the Bondi-Metzner-Sachs group

TL;DR

This paper reformulates the Bondi-Metzner-Sachs group using homogeneous projective coordinates, enabling linear representation of transformations and facilitating advanced geometric analysis of asymptotically flat spacetimes.

Contribution

It introduces a homogeneous projective coordinate framework for the BMS group, simplifying transformations and analyzing asymptotic symmetries in a linear setting.

Findings

Re-expressed BMS transformations in linear form

Derived explicit forms of asymptotic Killing vectors

Analyzed integral curves of supertranslation generators

Abstract

This paper studies the Bondi-Metzner-Sachs group in homogeneous projective coordinates, because it is then possible to write all transformations of such a group in a manifestly linear way. The 2-sphere metric, Bondi-Metzner-Sachs metric, asymptotic Killing vectors, generators of supertranslations, as well as boosts and rotations of Minkowski spacetime, are all re-expressed in homogeneous projective coordinates. Last, the integral curves of vector fields which generate supertranslations are evaluated in detail. This work prepares the ground for more advanced applications of the differential geometry of asymptotically flat spacetimes in projective coordinates.